On the Derivation of Logarithmic Corrections to Holographic Dark Energy

The phenomenological parameters α and β appearing in entropy-corrected holographic dark energy (ECHDE) are generally expected to be related to the quantum gravity coefficients γ and δ associated with logarithmic entropy corrections. However, the precise form of this relation can depend on how the dark energy density is derived from horizon thermodynamics. In this work, we consider two approaches: the first law of thermodynamics (dE = T dS) applied to a cosmological horizon, and a holographic scaling argument based on horizon degrees of freedom (ρ ∝ S / L 4 ). We find that, within the thermodynamic framework, logarithmic contributions do not explicitly appear under standard assumptions, whereas the holographic counting approach can retain such corrections more naturally. By comparing the two perspectives, we propose a possible mapping between (α, β) and (γ, δ), offering a tentative microscopic interpretation for these otherwise phenomenological parameters.